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Register a new userA proof of Saitohs conjecture for conjugate Hardy $H^{2}$ kernels
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Qian Guan
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In this article, we obtain a strict inequality between the conjugate Hardy $H^{2}$ kernels and the Bergman kernels on planar regular regions with $n>1$ boundary components, which is a conjecture of Saitoh.
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In this paper, we prove a conjecture posed by Li-Yang in cite{ly3}. We prove the following result: Let $f(z)$ be a nonconstant entire function, and let $a(z) otequivinfty, b(z) otequivinfty$ be two distinct small meromorphic functions of $f(z)$. If $f(z)$ and $f^{(k)}(z)$ share $a(z)$ and $b(z)$ IM. Then $f(z)equiv f^{(k)}(z)$, which confirms a conjecture due to Li and Yang (in Illinois J. Math. 44:349-362, 2000).
For any real $beta$ let $H^2_beta$ be the Hardy-Sobolev space on the unit disk $D$. $H^2_beta$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $beta>1/2$. In this paper, we study composition operators $C_varphi$ on $H^2_beta$ for $1/2<beta<1$. Our main result is that, for a non-constant analytic function $varphi:DtoD$, the operator $C_{varphi }$ has dense range in $H_{beta }^{2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $varphi(D)$. It follows that if the range of $C_{varphi }$ is dense in $H_{beta }^{2}$, then $varphi $ is a weak-star generator of $H^{infty}$. Note that this conclusion is false for the classical Dirichlet space $mathfrak{D}$. We also characterize Fredholm composition operators on $H^{2}_{beta }$.
We set a framework for the study of Hardy spaces inherited by complements of analytic hypersurfaces in domains with a prior Hardy space structure. The inherited structure is a filtration, various aspects of which are studied in specific settings. For punctured planar domains, we prove a generalization of a famous rigidity lemma of Kerzman and Stein. A stabilization phenomenon is observed for egg domains. Finally, using proper holomorphic maps, we derive a filtration of Hardy spaces for certain power-generalized Hartogs triangles, although these domains fall outside the scope of the original framework.
A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$. In this paper, we prove this conjecture for large $n$.
A sequence which is a finite union of interpolating sequences for $H^infty$ have turned out to be especially important in the study of Bergman spaces. The Blaschke products $B(z)$ with such zero sequences have been shown to be exactly those such that the multiplication $f mapsto fB$ defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also provide a particularly simple new proof of one of the known characterizations. One of the new characterizations is that they are interpolating sequences for a more general interpolation problem.
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